Methods for Project Evaluation

March 8, 2004

3/8/04

Nuclear Energy Economics and

Policy Analysis

1

Alternativ e Methods

• Present w orth (PW ) m ethod

• Future worth (FW ) method

• Annual worth (AW) method

• Benefit-cost ratio (BC) method

• Internal rate of return (IRR) method

3/8/04

Nuclear Energy Economics and

Policy Analysis

2

A ssum ptions

• Future cash flow s are know n w ith certainty

• A nalysis is in constant dollars

• Cost of capital is known

• Capital is always available for profitable projects (i.e, access to capital is not restricted.)

3/8/04

Nuclear Energy Economics and

Policy Analysis

3

P W Method :

NPV   n

N r  c

n

n  1

( 1  i ) n

Decision criterion: Accept if N PV>0; reject if NPV<0

N

F W Method :

FV  ( r  c )( 1  i )



n  0

N  n

n n

Decision criterion: Accept if FV>0, et c.

3/8/04

Nuclear Energy Economics and

Policy Analysis

4

Example of PW Method: Pricing a Bond

• At what price should a buyer purchase a 10-year bond paying 6% per year (payable semi-annually) that is redeemable at par value if the buyer is seeking a 10% per year yield? The face value of the bond is

$1000.

N = 10 x 2 = 20 periods

r = 6%/2 = 3% per period

i = [1.1 1/2 -1]100 = 4.9% per compounding period C = Z = $1000

V ( N )  $ 1000  ( P / F , 0.049,20)  $100 0 (0.03 )( P / A , 0.049, 20 )

 384. 1  377. 06  $761.16

Note : The yield typically increases for longer-term bonds.

3/8/04

Nuclear Energy Economics and

Policy Analysis

5

Example : Influenc e o f interes t rate s o n bon d prices

• A 10-year U.S. treasury bond that matures in eight years has a face value of $10,000. The bond pays 8% per year (payable quarterly). A prospective buyer of the bond wants to earn 10% per year on her investment (compounded quarterly) because interest rates have risen since the bond was issued. How much should the buyer pay for the bond?

V ( N )  $10 , 000  (0 . 02)  ( P / A , 0 .025,32)  $ 1 0 , 000( P / F , 0 . 025 , 32 )

 $8 , 907

. I.e., an increase in interest rates causes bond prices to decline.

3/8/04

Nuclear Energy Economics and

Policy Analysis

6

Example: Pricing stock

 Stock in a company represents a share of ownership, as opposed to a bond, which is essentially a promissory note.

 Common stock is more difficult to value than bonds because dividends and prices of common

stocks are not constant; investors hope that they will increase over time.

 If r eliable forecasts of future earnings, div idends, and s tock prices could be made, stock valuation would result from discounting the forecast cash f low.

Example (from Riggs and We st):

An investor is investigating the stock performance of two companies: A and B. Company A has consistently paid dividends that increase 10 cents per yea r while the selling price of the stock has averaged a 2% annual rise. Company B i s a fa st growing new company that has paid no dividends because all earnings are retained for expansion, but its market price is e xpected to increase by $10 per year. Current data about the two companies are su mmarized below:

Disregarding tax effects and brokerage commissions to buy or sell, which stock is more attractively

priced?

3/8/04

Nuclear Energy Economics and

Policy Analysis

7

Annual Worth (AW) Method

Example : An investment company is considering building a 2 5-unit apartment complex in a growing to wn. Because of the long-term growth potential of the to wn, it is felt that the company could average 90% of full occupancy for the complex each year. If the following items are reasonably accurate estimates, use the AW method to d etermine the minimum monthly rent tha t should be charged if a 12% rate of return per year is desired.

Land investment cost = $50,000 Building investment cost = $225,000 Study period, N = 20 years

Rent per unit per month = ???

Upkeep expense per unit per month = $35

Property taxes and insurance per year = 10% of total initial investment Assume: Land cost can be recovered at the end of the 20 year period

Solution : First determine the equivalent A W of all costs at an interest rate o f 12%/yr. To earn 12% on this project, the annual rental income must equal the AW of the costs:

Initial investment cost = $50,000 + $225,000 Taxes and insurance/yr = 0.1 x 275000 = $27,500 Upkeep/yr = $35 (12 x 25)(0.9) = $9450

Annual worth of capital costs = $275,000 ( A/P,0.12,20) - $50,000 (A/F, 0.12, 2 0)

= $36,123

Equivalent annual worth of costs = -$27000 - $9450 - $36123 = -$73073

Therefore, the minimum annual rental required equals $73,073 to achieve a 12% rate of return, and with annual compounding, the monthly rental amount is given by:

( 12  25)(0.9)  $ 270. 64

73,073

3/8/04

Nuclear Energy Economics and

Policy Analysis

8

Internal Rate of Return (IRR) Method

For a project with net cash flows, F j the IRR, i*, is given by

PV  i *   

N F

j

j  0  1  i * 

j

 0

Decisio n criterion :

If the m inimum required rate of return < i *, , accept the project. If the m inimum required rate of return > i * , reject the project.

3/8/04

Nuclear Energy Economics and

Policy Analysis

9

IRR Method (contd)

The equation for the IRR is an N th order polynomial in i*. There will i n general be more than one root. If more than on e of the roots is real and positive, how do we i nterpret the results?

Question: When is there a unique solution to the I RR problem?

If we write 1/(1+i*) = X, we can rewrite the IRR equation as

F 0 + F 1 X + F 2 X 2 + + F N X N = 0

No

Unique IRR; accept if > than minimum acceptable rate of return

Sign change in F’s >1?

Yes

Reject

Yes

More than one solution?

No

3/8/04

Nuclear Energy Economics and

Policy Analysis

10

Descartes ’ Rul e o f Signs :

For an N-th degree polynomial with real coefficients, the number of real, positive roots is never greater than th e number of sign changes in the sequence of coefficients.

The Project Balance, PB n

(the amount of money committed to a project at time n)

A n importan t distinction :

Projects for which PB(i * ) n  0 for all n < N

“PURE INVESTMENT” PROJECTS

Projects for which PB(i * ) n > 0 for some n

“MIXED INVESTMENT” PROJECTS

3/8/04

Nuclear Energy Economics and

Policy Analysis

11

3/8/04

Nuclear Energy Economics and

Policy Analysis

12

0 1 2 3

Single p ayment, pr oject 1

$58.73

$4 8 .73

0 1 2 3

Grad i ent s eries (decreasing),

p roject 3

$40

$50

0 1 2 3

Gradent s eries (increasing), p roject

0 1 2 3

Uniform s eries,

p roject 2

i = 10%

63.4

0 1 2 3

-100

-1 10 -121

63.4

0 1

3.67

3

-50.64

-100

63. 4

0 1

13.33

3

-41.27

-100

63 .4

0 1 2 3

-6

-60

-100

Sum mary

1. The PV, FW, and AW cr iteria always yield the same decision fo r a project

2. Only for pure investment projects is there a true IRR for the project.

3. For pure investments, the IRR and PV criteria yield identical acceptance/rejection decisions.

4. For mixed investments, the return on invested capital varies with the external cost of capital, and the I RR criterion isn’t meaningful. (The phenomenon of multiple IRRs can occur only with mixed investments, but even if there is only a single positive solution, it doesn’t necessarily provide useful information.)

5. The ag gregate B/C ratio criterion will al ways agree with the PV c riterion.

6. The pa yback period is not an a cceptable criterion taken on its own. In general it will not agree with the PV criterion. However, it may serve a useful purpose as a supplementary consideration.

3/8/04

Nuclear Energy Economics and

Policy Analysis

15